Fun and Games on a Chess Board Olga Radko November 19, 2017 I Names of squares on the chess board Color the following squares on the chessboard below: c3, c4, c5, c6, d5, e4, f3, f4, f5, f6 What letter do these squares form together? 1
II How many squares are there on a chessboard? A chessboard itself is a square with side 8. 1. The number of 1 1 squares on the chess board is. 2. What about bigger squares? Let's rst count squares of size 2 2: Idea: Instead of counting 2 2 squares, we will count the small 1 1 squares which can serve as the left lower corners of the 2 2 squares that t on the chessboard. First, shade the left lower corner of the 2 2 square above. 2
For each of the squares below, decide if it can be a left lower corner of a 2 2 square: Remember, the 2 x 2 squares on the chessboard can overlap. Like this: (a) square c3 Yes No (b) square g6 Yes No (c) square f8 Yes No (d) square h2 Yes No 3
Now color all 1 1 squares that can serve as the left lower corners of a 2 2 square: How many 2 2 squares can you t onto a chessboard? 4
3. For each of the squares below, decide if it can be a left lower corner of a 3 3 square: (a) square e6 Yes No (b) square g3 Yes No (c) square a7 Yes No (d) square f6 Yes No Now color all 1 1 squares that can serve as the left lower corners of a 3 3 square: How many 3 3 squares can you t onto a chessboard? 5
Now you can ll out the table below: Homework Count the number of 4 4, 5 5, 6 6 and 7 7 squares on the chessboard in the same way. In each case, use a chessboard picture to shade all the 1 1 squares that can be left lower corners of the bigger squares that t completely onto the chessboard. 6
Now color all 1 1 squares that can serve as the left lower corners of a 4 4 square: Color all 1 1 squares that can serve as the left lower corners of a 5 5 square: 7
Color all 1 1 squares that can serve as the left lower corners of a 6 6 square: Color all 1 1 squares that can serve as the left lower corners of a 7 7 square: 8
Fill out the table below with the numbers of squares: size of the square # of squares of this size 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 Now add up all the numbers in the right column to nd the total number of squares of all sizes. 9
II Rook Race Game Two players are playing the following game: Initial position: Two rooks are placed on two squares of a chessboard. Move: Move any of the rooks to the right by any number of squares. Goal: square. To be the last person to reach the rightmost 1. Play this game with your partner several times. Try to come up with a winning strategy. That is, come up with a method that allows you to win no matter what your opponent does. Only one of the players (rst or second) has a winning strategy. You need to nd it. Here are the initial positions: (a) Rook I on f3, Rook II on f6 (b) Rook I on d3, Rook II on d6 (c) In general, if both Rooks are the same number of squares away from the right edge, which player can win? How? 10
2. Now use the following initial positions: (a) Rook I on f3, Rook II on d6 (b) Rook I on e3, Rook II on a6 (c) Rook I on a3, Rook II on b6 (d) Rook I on c3, Rook II on g6 (e) In general, if the rooks are dierent number of squares away from the right edge, which player can win? How? (f) If you think you can handle any Rook race game, please challenge one of the instructors to play with you. The instructor will set up an initial position, and you will have a choice of being Player I or Player II. GOOD LUCK! 11
II Put Rook Into the Corner Game Two players are playing the following game: Initial position: One Rook is placed somewhere a chessboard. Move: Move the Rook down or left by any number of squares. Goal: To put the Rook into the left lower corner. 3. Play this game with your partner several times. Try to come up with a winning strategy. That is, come up with a method that allows you to win no matter what your opponent does. In every position, only one of the players (rst or second) has a winning strategy. You need to nd it. Here are the initial positions: (a) Rook I c3 Rook II on c5 (b) Rook on d4, Rook II on d6 (c) Rook on d4, Rook II on f7 (d) In general, if the Rook is on the diagonal connecting squares a1 and h8, which player can win? How? 12
(e) How does the game change if the Rook is placed away from the diagonal? Which player can win now? Homework: Play both the Rook Race and Put Rook into the corner games at home with your parents, friends, brothers and siblings. 13